Optimal order conditions for filter based regularization methods, with applications to variants of the Tikhonov method

TL;DR

This paper establishes optimal order conditions for filter-based regularization methods, including fractional and weighted Tikhonov variants, enhancing understanding of parameter choices for solving ill-posed linear problems.

Contribution

It derives new sufficient and necessary conditions for optimal regularization order, improving upon previous results for fractional and weighted Tikhonov methods.

Findings

Optimal order conditions for fractional Tikhonov method

Optimal order conditions for weighted Tikhonov method

Stronger conditions than previous literature

Abstract

We study filter based regularization methods for linear ill-posed problems between Hilbert spaces. We derive optimal order conditions under a-priori choice rules for the regularization parameter. Such analysis is applied to the fractional Tikhonov method and the weighted Tikhonov method recently investigated in [9] and [8], respectively. These variants of the classical Tikhonov method can be viewed as classes of methods depending on a further parameter, controlling the smoothness of the computed solution. From our previous analysis, we derive sufficient conditions on such parameter in order to have filter based regularization methods that agrees with the previous results in the literature. On the other hand, our analysis shows sufficient and necessary conditions to have an optimal order fractional Tikhonov method, which are stronger than those in [9]. The same analysis is performed also for the weighted Tikhonov method.